1.3. Quantum Physics Basics¶

1.3.1. Time Independent Two Level Systems¶

In general a quantum two level system can be described using Schrodinger equation

\[\begin{split}i\partial_t \begin{pmatrix} \psi_1\\ \psi_2 \end{pmatrix}= \frac{1}{2} \begin{pmatrix} -\omega_0 & 0 \\ 0 & \omega_0 \end{pmatrix} \begin{pmatrix} \psi_1\\ \psi_2 \end{pmatrix},\end{split}\]

where \(\omega_0\) is real. We define the unperturbed Hamiltonian,

\[\begin{split}H_0 = \frac{1}{2}\begin{pmatrix} -\omega_0 & 0 \\ 0 & \omega_0 \end{pmatrix}.\end{split}\]

Introduce perturbation

\[\begin{split}W = \frac{1}{2}\begin{pmatrix} -\delta & w \\ w* & \delta \end{pmatrix},\end{split}\]

where and identity matrix could have been added to restore the actual potential field. The system Hamiltonian becomes

\[\begin{split}H =& \frac{1}{2}\begin{pmatrix} -\omega_0 - \delta & w \\ w^* & \omega_0 +\delta \end{pmatrix} \\ =& \frac{1}{2}\begin{pmatrix} -\omega & w \\ w^* & \omega \end{pmatrix},\end{split}\]

where \(\omega = \omega_0 +\delta\) is called detuning and \(w\) is the Rabi frequency. The Schrodinger equation can be solved which gives us the general solution

\[\begin{split}\begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} = C_+ e^{\lambda_+ t}\eta_+ + C_- e^{\lambda_- t} \eta_-,\end{split}\]

where

\[\begin{split}\lambda_\pm =& \pm \frac{i}{2}\sqrt{w^2+\omega^2} \equiv \pm i \lambda_R\\ \eta_\pm =& \begin{pmatrix} \frac{-\omega \pm \sqrt{w^2+\omega^2}}{w^*} \\ 1 \end{pmatrix}.\end{split}\]

Rabi Frequency

Rabi frequency \(w\) will be generalized to the generalized Rabi frequency

\[R=\sqrt{w^2+\omega^2},\]

at which frequency the final system is oscillating.

We can determine the coefficients \(C_\pm\) using initial condition. Suppose we have the system initially in state

\[\begin{split}\begin{pmatrix} 1\\ 0 \end{pmatrix}.\end{split}\]

The coefficients are

\[C_\pm = \pm \frac{1}{2} \frac{w^*}{\sqrt{w^2+\omega^2}}.\]

The final solution is

\[\begin{split}\begin{pmatrix} \psi_1(t) \\ \psi_2(t) \end{pmatrix} = \frac{w^*}{\sqrt{w^2+\omega^2}} \begin{pmatrix} -i\frac{\omega}{w^*} \sin(\lambda_R t) + \frac{\sqrt{\omega^2+w^2}}{w^*} \cos(\lambda_R t)\\ i \sin(\lambda_R t) \end{pmatrix}.\end{split}\]

The transition probability is

\[\begin{split}P_{1\to 2} =& \frac{w^2}{w^2+\omega^2} \sin^2(\lambda_R t)\\ =& \frac{w^2}{w^2+\omega^2} \sin^2 \left(\sqrt{w^2+\omega^2}/2 \right).\end{split}\]

1.3.1.1. Energy Spectrum¶

It’s important to visualize the energy spectrum. The energy levels are in fact

\[E_\pm = \pm \lambda_R.\]

We notice that for zero off-diagonal perturbation \(w=0\), we have

\[E_\pm = \pm \omega,\]

which means the energy levels are linear to \(\omega\). At \(\omega=0\) we have degeneracy of the two state, i.e., the two states have the same energy. However, as we introduce non-zero off-diagonal perturbation \(w\neq 0\), the degeneracy is gone. The crossing of the energies is avoided.

1.3.1.2. References of Rabi System¶

A very short and concise introduction: Rabi Model by Michael G. Moore at MSU.

1.3.2. Time Dependent Two Level System¶

The previous calculations are for time independent perturbations. For time dependent perturbations, the situation would be drastically different.

Here we introduce a system with Hamiltonian

(1.4)¶\[\begin{split}H = \frac{1}{2}\begin{pmatrix} -\omega_0 & w e^{i k t} \\ w e^{-ikt} & \omega_0 \end{pmatrix} .\end{split}\]

To solve the Schrodinger equation, we can go to the corotating frame. The solution to it is very similar to time independent case, with a detuning shifted by \(-k\) instead of \(\delta\). In other words, in the time dependent Rabi system \(-k\) is equivalent to \(\delta\) in the time independent system. Thus the detuning becomes

\[\omega = \omega_0 - k.\]

1.3.2.1. References of Time Dependent Rabi System¶

Rabi Oscillations by Thilo Bauch and Goran Johansson.

1.3.3. Bloch Sphere and Bloch Vectors¶

A two level system density matrix can be expanded using identity matrix and Pauli matrix,

\[\rho = \frac{1}{2}(1 + \mathbf P \cdot \boldsymbol \sigma),\]

as defined in Neutrino Flavour Isospin. The vector \(\mathbf P\) represents the vector of state in Bloch sphere.

We can also project the Hamiltonian into this space,

\[H = - \frac{\boldsymbol \sigma}{2} \cdot \mathbf H.\]

We take the (1.4) as an example. A simple derivation shows that the equation of motion for the Bloch vector or the flavor isospin vector is precession

\[\dot {\mathbf P} = \mathbf P \times \mathbf H,\]

where

\[\begin{split}\mathbf H = \begin{pmatrix} -w \cos(kt)\\ w\sin(kt)\\ \omega_0 \end{pmatrix}.\end{split}\]

We identify this vector as a rotation around z axis. It also becomes obvious that the so called time indepedent case is literally (1.4) in a corotating frame.